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### Mapping a Sphere to a Plane

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Date: 11/28/2001 at 12:15:18
From: Candice
Subject: Maps of the world

Maps of the world are always distorted in some way when put on a flat

I have to answer this question on a college level and an elementary
level. I would explain this to elementary students with an orange
peel. We can see that an orange peel covers the whole orange before
we peel it. Then after we peel it we can see that the peel would not
lie flat on a table. I 'm not sure where to go from here.

I'm not sure which facts of geometry I should use to investigate this
on a college level. I think that using the formula of the surface
area of a sphere might work, but I'm not sure how to use this formula

Thanks,
Candice
```

```
Date: 11/28/2001 at 13:18:31
From: Doctor Tom
Subject: Re: Maps of the world

Hi Candice,

The proof that it's impossible uses differential geometry, but I can
perhaps indicate the idea behind the proof.

Any smooth surface has a "curvature" at every point. In the case of
the plane, the curvature happens to be zero everywhere. The curvature
may not be exactly what you think it is. On a cone, away from the tip,
the curvature is also zero. (There are different sorts of curvature,
but the type I'm interested in has zero curvature.) And, in fact, a
cone could be slit and flattened out on a plane.

You can locally measure this curvature if you "live" on such a surface
as follows: Take a point and put a nail in the ground. Take a piece of
string of length 1 tied to the nail, and draw a "circle" by connecting
the other end to a pencil and dragging it around as far from the nail
as possible. Now measure the length of the circle. If it's 2*pi, the
curvature is zero. If it's bigger, the curvature is negative. If it's
smaller, the curvature is positive.

A sphere obviously has positive curvature everywhere since the circle
on the surface of the sphere is smaller than it would be on a plane.
(If you have trouble imagining this, think of a 1-foot string on a
2-foot sphere.) Things like saddles are the opposite - the circle on a
saddle would be bigger than on a plane.

If you try to imagine mapping a sphere to a plane in a way that
preserves all lengths, you're obviously out of luck. The the points on
the plane will all have to be the same distance away from the image of
the center, but then you will have changed the lengths around the
circle, so the scaling is wrong there.

You can map a small enough piece of a sphere to the plane with an
arbitrarily small error, but as you decrease the allowable error, the
map gets smaller and smaller, and none of them are mathematically
perfect.

- Doctor Tom, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Higher-Dimensional Geometry
High School Higher-Dimensional Geometry

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